Min-max model for the network reduction problem

Transcription

1 Min-max model for the network reduction problem tefan PE KO University of šilina, Slovakia Mathematical Methods in Economics - Jihlava, Czech Republic September 2013

2 Network reduction problem We deal with solving a N P-hard problem that occurs in reduction of transport networks. The reduction means to omit some edges. There are two requirements imposed on the resulting reduced network: 1 The upper bound of the total length of reduced network is known. 2 The distance between arbitrary two points in reduced network should not exceed q-multiple of that in original network, where q should be as small as possible. There are many applications of just mentioned problem, e.g. when designing a bus line system in a town. We have found interesting application in biology Hassan (1992), where the acyclic network considered there has two attributes in each arc: time and length, with a time limit on the shortest length path.

3 Dierent network reduction problems ƒerná, ƒerný, Pe²ko, Czimmermann (2007) studied the family of subgraphs not lengthening some important trip more that for the given percentage. P ibyl (2009) deals with a single bus route design problem where a goal of reduction of network is to minimize the mean walking distance of passengers to the nearest stops of bus route. Czimmermann (2010) studied the computational complexity of the problem Admissible Lengthening of Important Routes (ALIR) and showed that problem is N P-hard for any lengthening parameter q>1. Michali ková, Jáno²íková (2013) deals with exact and heuristic IP solving the problems of the ALIR for bus lines system. Pe²ko (2013) studied the programming approach for the network reduction problem for small instances with xed q.

4 Mathematical formulation We are given a weighted graph G = (V, E, d) and a number T, T > 0. The goal is to nd number q and a connected spanning subgraph G q = (V, E q, d q ) of G with minimal value of parameter q, q 1 such that the length of subgraph G q is at most T and the distance between each pair of vertices of removed edge is at most q-times of the length of this edge. q min (1) d(e) T, (2) e Eq d q (u, v) = d(u, v) {u, v} E q, (3) d q (u, v) q d(u, u) {u, v} E E q, (4) E q E, (5) q 1. (6) where d(u, v) resp. d q (u, v) is length of shortest u v path in graph G resp. G q.

5 Illustrative example Dashed edges pictured can be removed from original network Figure: Original G and reduced G q transportation networks with q = 1.4.

6 LP model for distance matrix Goal is to nd a distance matrix X = (x ij ) where x ij is length of the shortest i j path in graph G = (V, E, d). For any connected graph G is possible solve this as following LP problem: x ij max i V j V :i<j x ik + x kj x ij (i, j, k) V V V : i j k i x ij = x ji i V, j V : i < j x ij d ij {i, j} E. Length of shortest path are here modeled as a sum of lower bounds of lengths of its edges.

7 CP model for distance matrix in MiniZinc Parameters nvertix number of vertices, nedge number of edges, innity satisfactory big integer number, V = {1,..., nvertix} set of vertices, E = {1,..., nedge} set of order of edges, Edges[] matrix of weighted edges of size nedge 3 where k-th edge {i, j} weighted by integer d ij is represented by k-th row with Edges[k, 1] = i, Edges[k, 2] = j, Edges[k, 3] = d ij. Variables X [] matrix of distances of size nvertix nvertix with domain {0,..., innity}, total_cost sum of all distances is maximized.

8 Constraints for distance matrix in MiniZinc C1: Since x ij d ij and x ij = x ji for each edges {i, j} we can write in MiniZinc: forall(e in E)( X[Edges[e,1], Edges[e,2]] <= Edges[e,3] /\ X[Edges[e,1], Edges[e,2]] = X[Edges[e,2], Edges[e,1]] ); C2: The feature of the distance matrix is that it is a metric on the set V and so can write: forall(i in V, j in V, k in V where i!=j /\ k!=i /\ k!=j)( X[i,k] + X[k,j] >= X[i,j] );

9 MiniZinc model for min-max network reduction Model for distance matrix is a base for network reduction model. Parameters (continue) T upper bound of length of reduced network. Variables (continue) q100 integer coecient of extension is equal 100 q Y binary matrix of size nedge where Y [k] = 1 if k-th edge is in reduced network and Y [k] = 0 otherwise, Z matrix of ows of size nedge 2 where Z[k, 1] is ow Edges[k, 1] Edges[k, 2] and Z[k, 2] is ow Edges[k, 2] Edges[k, 1]. Interpretation of the object function is changed: total_cost q100 max.

10 Reduction s C3, C4, C5 C3: The maximum length of reduced network is given : sum(e in E)(Edges[e,3]*Y[e]) <= T; C4: In reduced network we choose some edges only: % upper bounds and symmetry forall(e in E) ( X[Edges[e,1],Edges[e,2]]<=max(Edges[e,3],(1-Y[e])*infini /\ X[Edges[e,1],Edges[e,2]]=X[Edges[e,2],Edges[e,1]] ); C5: Constraint C2 above triangulation inequality remains valid: forall(i in V, j in V, k in V where i!=j /\ k!=i /\ k!=j)( X[i,k] + X[k,j] >= X[i,j] );

11 Reduction C5, C6 C5: A feasible reduction of edge {i, j} is possible only if exists a vertex k so that x ij = x ik + x kj : forall(e in E) ( Y[e] = 0 -> exists(k in V diff {Edges[e,1],Edges[e,2]}) X[Edges[e,1],Edges[e,2]]=X[Edges[e,1],k]+X[k,Edges[e,2]] ); C6: The feasible extension of shortest i-j path is possible: forall(e in E) ( Y[e]=1->X[Edges[e,1],Edges[e,2]]=Edges[e,3] /\ Y[e]=0->X[Edges[e,1],Edges[e,2]]<=q100*Edges[e,3] div 100 /\ Edges[e,3]<infinity->X[Edges[e,1],Edges[e,2]]>=Edges[e,3] );

12 Reduction C7, C8 C7: The reduced network is at least spanning tree: sum(e in E) (Y[e]) >= nvertex-1; C8 The spanning tree is constructed using ows from every (non-root) vertex to root 1: sum(e in E)(bool2int(Edges[e,1]=1)*Y[e]*Z[e,1])=nVertex-1 /\ forall(i in 2..nVertex) ( sum(e in E) (bool2int(edges[e,1]=i)*y[e]*z[e,1] +bool2int(edges[e,2]=i)*y[e]*z[e,2]) = sum(e in E) (bool2int(edges[e,2]=i)*y[e]*z[e,2] +bool2int(edges[e,1]=i)*y[e]*z[e,2])-1 );

13 Computer experiments cpu MHz , cache size 6144 KB, 2 cores, T Tq V E G [mm:sec] G q [mm:sec] q T [%] :18 0: :13 0: :23 22: :32 13: :34 34: :33 102: :35 65: :41 110: :27 173: Table: Runtimes for randomly generated planar instances G with a xed limit T = 1000 and the calculated length T q of reduced network G q. T q = e Eq d(e)

14 Thank you for your attention...